text: It's straightforward to show that for two-dimensional plane flow  satisfies the curl-divergence equation
  where  is the  rotation matrix corresponding to a  anticlockwise rotation about the positive  axis. This equation holds regardless of whether or not the flow is incompressible.
  If the flow is incompressible (i.e., ), then the curl-divergence equation gives
  Then by Stokes' theorem the line integral of  over every closed loop vanishes
  Hence, the line integral of  is path-independent. Finally, by the converse of the gradient theorem, a scalar function  exists such that
  Here  represents the stream function.
  Conversely, if the stream function exists, then . Substituting this result into the curl-divergence equation yields  (i.e., the flow is incompressible).
  In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.